CONVECTIVE DIFFUSION PROCESS OF BLOOD VESSELS IN THE PRESENCE OF POROUS EFFECTS

Anil Kumar

Department of Mathematics Dr. K. N. Modi Institute of Engineering & Technology,

Modinagar-201204, Ghaziabad (India) drkumaranil73@yahoo.co.in ABSTRACT

In this paper, convective dominated diffusion process in an axi-

symmetric tube with a local constriction simulating a stenosed artery

considering the porosity effects is carried out. The investigations demonstrate

the effects of wall shear stress and recirculating flow on the concentration

distribution in the vessels lumen and on wall mass transfer keeping the

porosity in view. The flow is governed by the incompressible Navier-Stokes

equations for Newtonian fluid in porous effects. The convection diffusion

equation is used for the mass transport. The effect of porosity is examined on

the velocity field and wall shear stress. The numerical solutions of the flow

equations and the coupled mass transport equations are solved using finite

difference scheme. The study explains the effect of porosity of the flow on the

mass transport. The works that consider the possibility of reducing these flow

instabilities using porous effect are reviewed.

Key words: Blood Vessels, Porosity, Wall Shear Stress, Simulation, Convection

Diffusion

1. INTRODUCTION

Computational fluid dynamics simulations of complex flows in vascular

passages such as cerebral or carotid arteries can provide clinicians with

information needed to evaluate how pathologies from, how they evolve and

ultimately how they are effectively treated. The geometry of blood vessels and

arterial wall, their structure and mechanical properties depend on the pressure

and flow conditions. In general stenosed arteries respond to sustained change

in pressure of flow with remodeling functional characteristics of the artery. The

porosity effects on blood flow cannot be ignored while studying a realistic

problem. Some of the studies on the arterial flows are given as follows.

Analytical transport phenomena are important to the understanding of vascular

diseases is established by Fry and Vaishnav (1980).

Liepsch and Moravec (1984) investigated the flow of a shear thining

blood, analog fluid in pulsatile flow through arterial branch model and observed

large differences in velocity profiles relative to these measured with Newtonian

fluids having the high shear rate viscosity of the analog fluid. Rodkiewicz et al.

(1990) used several different non-Newtonian models for simulations of blood

flow in large arteries and they observed that there is no effect of the yield stress

of blood on either the velocity profiles or the wall shear stress. Out of these a

large number of numerical studies concentrate on the transport phenomena in

the microcirculation. Bassingthwaitghte et al. (1992) studied the exchange of

gases between the systemic capillaries and the surrounding tissue. Very much

attention has not been given to the numerical investigation of transport

phenomena in large arteries keeping porosity in view.

The reason for this may be that most of these phenomena are highly

convection dominated due to the low diffusion coefficients in quite a large

number of cases. In the arterial system there is evidence that fluid dynamic

factors, especially shear forces, affect blood phase transport as well as

properties of the vascular wall like wall permeability to solutes of gases [Tarbell

(1993)]. Cavalcanti (1995) has discussed the inadequacies of such studies for

the determination of the model for plaque growth. These observations provide

future direction for research. Back et al. (1996) founded the important hemo-

dynamical characteristics like the wall shear stress, pressure drop and

frictional resistance in catheterized coronary arteries under normal as well as

the pathological conditions due to stenosis being present. Rapptisch and

Perktold (1996) investigated computer simulations of convective diffusion

process in large arteries. For heat transfer in muscle tissues, Weinbaun et al.

(1997) found that a correction factor of efficiency needs to be multiplied by the

perfusion source term in the Pennes equation for bio heat transfer in a muscle

tissue. This coefficient is a function of vascular cross-section geometry and is

independent of the Peclet number

Korenga et al. (1998) consider a biochemical factors such as gene

expression and albumin transport in atherogenessis and in plaque rupture.

These have been shown to activate by hemo-dynamic factors in wall shear

stress. However the concept of suction by tissues present in the vessels has not

been considered in these studies. Rachev et al. (1998) have considered a model

for geometric and mechanical adaptation of arteries. Lei et al. (1998)

investigated a complex model for the transvascular exchange and extravascular

transport of both fluid and macromolecules in a spherical solid tumor, the

micro vascular lymphatic and tissue space were each considered as a porous

medium. Tang et al. (1999) analysed blood flow in carotid arteries with stenosis.

Berger and Jou (2000) measured wall shear stress down stream of axi-

symmetric stenoses in the presence of hemo-dynamics forces acting on the

plaque, which may be responsible for plaque rupture. Stroud et al. (2000)

founded the influence of stenosis morphology on flow fields and on quantities

such as wall shear stress among stenotic vessels with very mild stenosis.

Sharma et al. (2001) considered a mathematical analysis of blood flow through

arteries using finite element Galerkin approaches. Tada and Tarbell (2000)

utilized the Brinkman model to investigate the two-dimensional interstitial flow

through the tunica media of an artery wall in the presence of an internal elastic

lamina. Sharma et al. (2001) profounded MHD flow in stenosed artery using

finite difference technique. Khaled and Vafai (2003) studied flow and heat

transport in porous media using mass diffusion and different convective flow

models such as Darcy and the Brinkman models, Energy transport in tissue is

also analysed.

In the present study the steady convective diffusion of oxygen in an axi-

symmetric tube with a local constriction simulating a stenosed blood vessel is

investigated numerically applying a passive transport law for the flow of and

mass transfer parameter with effect of porous to take account of wall porosity.

An important aspect of this analysis is the comparison of two models for the

solute wall flux one is assuming constant wall permeability and the second

incorporating shear dependant wall permeability in the presence of porosity.

This work was motivated by the study of lateral dispersion of cholesterol in

arteries, where the blood containing cholesterol flows through and past the

porous medium [Shivkumar (2001)]. Sharma et al. (2004) studied performance

modeling and analysis of blood flow in elastic arteries. Kumar et al. (2005)

studied computational technique for flow in blood vessels with porous effects.

2. Formulation of the Problem:

In the present investigation the flow formulation for Navier-Stokes

equations for incompressible Newtonian fluids in the porous effect has been

considered. For an axi-symmetric coordinate (z, r) the equations can be written

as:

,112

2

2

2

ku

ru

rru

zuv

zu

ru

zuu

tu ν

ρν −⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

(1)

kv

rrrrzv

rrzu

tνννννν

ρνννν

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

22

2

2

2 11 (2)

0=+∂∂

+∂∂

rrzu νν

(3)

where u and v are the axial (z) and radial (r ) velocity components, ρ is the

constant fluid density, p is the fluid pressure and ν is kinematic coefficient of

viscosity.

The stream function –viscosity form of the Navier-Stokes equation has a major

advantage over the primitive variable from in two-dimensional incompressible

flow problem so far as numerical solutions of these equations are concerned.

Here

.1,1zr

vrr

u∂∂

−=∂∂

=ψψ

and vorticity is

.ru

zv

∂∂

−∂∂

=ζ (4)

The equations (3) can be written as,

,12

2

2

2

rrr

rz ∂∂

=+∂∂

+∂∂ ψζψψ

(5)

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

∂∂

+∂∂

+∂∂

=∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

−∂∂

∂∂

+∂∂ ζζζψψνψζζψζψζ

krrrrzzrrzzrrt111

22

2

2

2

2 (6)

Taking the r-derivative of equation (1) and subtracting the z-derivative of

equation (2), pressure term has been eliminated.

r

Lu Ls Ld

L0/2

L0/4

z

Figure 1. Geometrical construction of an artery segment with stenosis.

The mass transport, which is coupled to the velocity field, is modeled by the

convection diffusion equation as given below:

,12

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

rc

rrc

zcD

rcv

zcu

tc

(7)

where c denotes the solute concentration and D the constant diffusion

coefficient. To make the parameters non –dimensional forms we put:

0* / Lzz = 0

* / Lrr = ,/ 0* Uuu =

,/ 0* Uvv = ,/ 0

* Ccc =0

0*

LUtt = (8)

The convective diffusion equation can be written in non-dimensional forms as:

,11.

*

*

*2*

*2

2*

*2

*

**

*

**

*

*

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

rc

rrc

zc

Prcv

zcu

tc

e

(9)

where the reference length, the reference velocity and the reference

concentration. Peclet number is defined as:

0L 0U 0C

.00cee SR

DLU

P == (10)

where µ

ρ 00 LURe = is Reynolds number and DSc ρµ /= is Schmidt number.

For the large Peclet number the hyperbolic part *

**

*

**

rc

zcu

∂∂

+∂∂ ν , of the equation

becomes dominant. The elliptic equations are used to generate the boundary

fitted coordinate system. We are consider the following transformation.

,0,0 2

2

2

2

2

2

2

2

=∂∂

+∂∂

=∂∂

+∂∂

rzrzηηξξ

(11)

These equations constitute a transformation form the physical plane to

numerical plane. This transformation is governed by the elliptic equation; it is

called elliptic grid generation [Thompson (1974)]. The relations between the

physical plane and the transformed plane are:

,0212

222

212

2

211

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂−

∂∂

ηξηηξξzSzRzazaza

J (12)

,0212

222

212

2

211

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂−

∂∂

ηξηηξξzSrRrarara

J (13)

The metric coefficient and the Jacobian of the transformation are:

+∂∂

=∂∂

+∂∂

= 2

212

2

2

2

211 ,

ξηηzarza ,2

2

ξ∂∂ r

,,22

ξηηξηξηξ ∂∂

∂∂

−∂∂

∂∂

=∂∂

∂∂

+∂∂

∂∂

=rzrzJrrzza

The governing equations can be written as a coupled set of equations in terms

of the stream function and vorticity as given below:

,212

222

212

2

211

2 ψζηψ

ξψ

ηψ

ηξψ

ξψ SrSRaaa

J=+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂−

∂∂

(14)

,2112

222

212

2

211

2 ζζξηξη

ηζ

ξζ

ηζ

ηξζ

ξζ

ηζν

ξζζ KSSRaaa

Ju

rt−=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂−

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

(15)

where

,1,1⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

−=ηψ

ξξψ

ην

ηψ

ξξψ

ηrr

Jrzz

Jru

,1,1,1⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ηψ

ξξψ

ηηηνν

ηη ψξηξη zz

JrSruz

Jzvru

Ju

} ,12⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

−−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

=rv

rzzvrr

JrS ζζ

ηζ

ξξζ

ηζ

ηψ

ξξψ

ηζ

,⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

−=ηψ

ξξψ

ηηψ

ξξψ

ην

ζrrzz

JrK

2.1 Boundary Conditions

Assuming a steady flow a fully developed parabolic velocity profile at the inlet

are:

( ) ,0],)2(1[2 2

00 =−= z

LrUru (16)

at the wall a no slip condition is:

u = v = 0

and at axis of symmetry the normal velocity gradient is zero:

.0,0,0 =∂∂

=∂∂

=rr

u νν

At the out flow boundary section zero surface traction force is assumed to be

zero i.e.

,0=∂∂

+−nun µp

where n is the outward unit normal vector and u=(u, v).

Evidently no slip conditions as given below:

0,0 =∂∂

=∂∂

zzζψ

at ,1=ξ (17)

and ,5.0=ψ 0=∂∂

∂∂

+∂∂

∂∂

ηψ

ξξψ

ηrr

at ,1=η

Following Rapptisch and Perktold (1996) the boundary conditions for the

transport diffusive equations are constant concentration at the inlet,

c= , (18) 0C

Zero axial concentration gradient 0=∂∂zc

at the outlet condition,

And for symmetry condition, 0=∂∂rc

,

For the diffusive flux at the wall, we consider the first model with constant wall

permeability,

,wwall

w cncDq α=∂∂

−= (19)

where α is constant wall permeability and is the wall concentration.

And the other model permeability is dependent on the wall shear stress are

given by,

wc

( ) ,nnwall

w cgncDq τ=∂∂

−= (20)

where ( ng τ ) is the shear dependent wall permeability function.

3. Numerical Method

The spatially discrete from of the governing equations is obtained using a finite

difference scheme.

The semi-discretized governing equations are written in the following from:

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

∆+

∆∆−−+

∆ jiji

jiji

jiji

jiDD

aDD

aDD

aj .2

22.

.00

12.

.2

11.

.2 21 ψ

ηψ

ηξψ

ξηηηξξξ

- ,22 ....0

..0

.jjijiji

jiji

ji SrDs

DR

ψηξ ζψ

ηψ

ξ=+⎟⎟

⎠

⎞⎜⎜⎝

⎛∆

+∆

(21)

⎟⎟⎠

⎞⎜⎜⎝

⎛∆

+∆

+∂∂

jiv

jiu

jiji DD

rt .0.0,

. 221 ζ

ηξηζ

ξξηζ ξξ

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∆

+∆

−⎟⎟⎠

⎞⎜⎜⎝

⎛∆

+∆∆

−∆ −++ ji

jiji

jiji

jiji

jiji

ji

jiD

SD

RDDaDDagDDa

j ..

.0.

.2.

22

.00.

12

.2.

11

.2 2221 ξ

ηξ

ξξ

ηξ

ηξξ

ξηξηηηξξξ

= (22) ,.. jiji KS ξξ −

for i=2,…m-1, j=2…,n-1.

The difference operators are defined as:

andFFFDFFFD jijijijijiji .1....1. , −−++ −=−= ξξjijiji FFFD .1.1.0 −+ −=

Similarly and 1....1.. , −−++ −=−= jijijijijiji fffDfffD ηη ,1.1..0 −+ −= jijiji fffDη

The discrete form of the boundary conditions given above can be obtained using

the finite difference scheme.

,0,,211 ...,1.,1

2,1

2,1 =−=⎟

⎠⎞

⎜⎝⎛ −= jmhjmhjmhjjijjj DDrDrrr ψψζψ ηζη (23)

,0.... =− jmhjmhjmhjmh DrDDrD ζζ ηζζη (24)

for J=2,…,n-1,

,01. =iψ ,5.0,0 .. == niji ψζ (25) ,0.... =+ nihnihnihnih DxDDxD ψψ ηξξη

for i=2,…,m-1,

where ,43,43 .2.1...2.1.. jijijijihjijijijih FFFFDFFFFD ++−− −+−=+−= ξξ

and (26) ,43 2.1... −− +−= jijijijih FFFFDη

while all the other boundary conditions are conventional, there is a

computational difficulty with the no slip condition, since no vorticity value is

available at the wall, j=n in order to solve the difference equations (22). For

computation purpose, the following iterative algorithm is used following Huang

et al. [1995]. This scheme is used for temporary discretization to ensure a

second order accurate solution both in time and space for both convective and

diffusion terms.

NUMERICAL ALGORITHM:

Step1. Solve equation (21) for j=2,…, n-1 and i=2,…, m-1. The stream function

)(ψ at j=n-1 is calculated using both no slip boundary condition (17) at the wall.

Gauss-Siedal method is used to solve the algebraic equations.

Step2. Update and using ξηµ ξηv ψ from the previous step.

Step3. To calculateζ at j =n-1 for i=1,…, m-1 use relation (21), coupling ψ and

ζ , is satisfied at j=n-1;

Step4. Solve equation (22) at j=2,…,n-1 for i=2,…,m-1 using the value at j=n-1

from step (3) and necessary conditions at the other boundaries. Again using

Gauss-Siedal method solves the algebraic equations.

Step5. Steps (1)-(4) are repeated until convergence is reached. The solution of

convective diffusion equation can be obtained using the finite element

technique following Rapptisch and Parktold(1996). The present algorithm is

economic and efficient having a sharp convergence.

4. Results and Discussion

The simulation of the steady convective diffusion of dissolved oxygen in stenotic

artery model has been performed and streamlines as well as continuous

contour are given in figure 2(A) and 2(B). The simulation parameters for the flow

and oxygen transport are such that the physiological conditions in the human

abdominal aorta match with the same. The geometric parameters are taken as:

inlet diameter L0 =1.42 tube length upstream of the constriction diameter

/2=0.71 cm., length of the down stream domain L =65.5 cm. For the flow

simulation and oxygen transport following parameters are applied: [Rappitsch

and perktold (1996)], The simulation parameters are[Osenberg (1991)], mean

flow rate =17.5 mls , mean inlet velocity, =10.52cms . Kinemetic

viscosity=0.333 poise.

0L d

1−0U 1−

The resulting Reynolds number = 450, =2.58eR 0C ×10 mlcm , D =1.63− 3− ×10

and reference wall flux

5−

12 −scm

mlgmmsmlcmg -50

1260 103.0 ,86 ,1082.4 ×=Η=Ρ×= −−− λ

( λ/mm 0013 Cgcm =ΡΗ −− ). The Peclet number Pe =930000 indicate highly

convection problem. The qualitative characteristic of the is concentration

contours in the separated flow region shows sharp concentration gradient near

the wall and the separating stream line are very similar to shape of the contours

published by Ma et al. (1994) for mass transfer in sudden expansion tube

model.

Figure 3(a) is drawn for the normalized mean cross sectional concentration. The

comparison of the results with shear dependent wall permeability and constant

wall permeability shown only slight difference. Figure 3 (b) depicts the main

differences between the two permeability models occurring in the wall

concentration. Figure 3(c) explains the non –dimensional diffusive wall flux.

This study may have an important impact on the understanding of development

of atherosclerotic diseases. The comparison of the result in the modal with

constant wall permeability and in a model with shear dependent wall

permeability demonstrates great differences in wall permeability. The results

show that taking into account shear dependent permeability characteristics,

depending on the special solute or gas, may have significant influence on the

concentration distribution near and at the wall and consequently on the wall

flux. The velocity profiles are shown in figure 4. Comparison between

experimental and numerical simulation are made for the pressure distribution

along the solid surface and the axial velocity profile.

5. Conclusion

Steady convective diffusion has studied on an axi-symmetric stenotic artery

model in the presence of porous effect. The numerical solution of the flow

equations is obtaining applying the finite differences scheme. The numerical

result indicate strong dependence of oxygen transport to the wall in the

geometric flow region by influencing the concentration boundary layer, which,

depends on the local fluid mechanical shear. In the model with shear stress

dependent permeability wall shear stress not only affects the concentration

boundary layer, but also vascular permeability. In the shear dependent

permeability model a second flux minimum occurred at the down stream end of

the recirculation zone where wall shear stress and therefore wall permeability is

slow. Numerical simulation of analytical stenosis offers a non-invasive means of

obtaining detailed flow patterns associated with the disease. The effect of the

porous medium is an elegant device for flow control. It is found that models for

convective mass transport through porous media are widely applicable in the

production of the osteoinductive material, simulation of blood flow of tumors

and muscles and in modeling blood flow when fatty plagues of cholesterol and

artery clogging clots and formal in the lumen.

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50 Shear dependent wall

45 permeability

40 Constant wall permeability

35

30

25

20

15

10

5

0 0 2 4 6 8 10 13 15 17 19 25 35 45 z / L0

Fig.3 (a) Normalized mean cross sectional concentration

60 Shear dependent wall permeability

50 Constant wall permeability

40

30

20

10

0

0 4 6 8 10 12 14 16 18 20 25 35

z / L0

Figure 3 (b): Normalized wall concentration non-dimensional wall flux

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9 10 15 20 25 30 35 40 45 50z/L0

q/q 0

Shear Dependent WallPermeabilityConstant WallPermeability

Figure 3 (c): Normalized Wall Concentration non-dimensional Diffusive flux.

1.0 0.55 r 0.0 -0.55 -10 Z =-2.01 Z =0 Z =2.01 Z =4.01 Z =6.01 Z =8.01

Figure 4 (a) Variation of axial velocity profile with porosity (K=0.1). 1.0 0.55 r 0.0 -0.55 -10

Z=-2.01 Z =0 Z =2.01 Z =4.01 Z =6.01 Z =8.01

Figure 4 (b): Variation of axial velocity profile with porosity (K=0.2).